In the Eternal Forest, the trees are perfectly circular, each having a diameter of exactly one meter. They are arranged in a flat, infinite rectangular grid. The center of each tree is ten meters away from the centers of each of its closest neighbors.

There are many paths through the Eternal Forest. Each path is infinite in length, constant in width, and perfectly straight. Trees don't grow on the paths, but every path will have tree trunks touching it on either side.

What is the narrowest possible path through the Eternal Forest?

I checked every possible numerator for slopes and found the following positive answers:

1/n: see previous post

Slope     Width

2/3     1.7735
2/5     .85695
2/7     .37361
2/9     .08465
3/4     1
3/5     .71499
3/7     .31306
3/8     .17041
4/5     .56174
4/7     .24035
4/9     .015346165134
5/6     .28037
5/7     .16248
5/8     .06000
6/7     .08465

all other fraction slopes have no positive path.      

Posted by Jer on 2005-05-27 19:08:52